Derivation of soil screening values for metals in Portuguese natural soil

The increasing human activity has been responsible by profound changes and a constinuos degradation of the soil compartment in all the European territory. Some European policies are appearing focusing soil’s protection and the management of contaminated sites, in order to recover land for other uses. To regulate the risk assessment and the management of contaminated soils, many European member states adopted soil guideline values, as for example soil screnning values (SSV).These values are particularly useful for the the first tier of the Ecological Risk Assessment (ERA) processes of contaminated sites,especially for a first screening of sites requiring a more site-specific evaluation. Hence, the approriate definition of regional SSVs will have relevant economic impacts in the management of contaminated sites. Portugal is one of European Member States that still lack these soil guideline values. In this context, this study gaves a remarkable contribution in the generation of ecotoxicological data for soil microbiological parameters, terrestrial plants and invertebrates for the derivation of SSVs for uranium (U), cadmium (Cd) and copper (Cu), using a Portuguese natural soil, representative of a dominant type of soil in the Portuguese territory. SSVs were derived based on two methods proposed by the the Technical Guidance Document for Risk Assessment of the European Commission; namely the assessment factor method (AF) and the species sensitivity distribution (SSD) method (with some adaptations). The outputs of both methods were compared and discussed. Further, this study laid the foundation for a deeper reflection about the cut-off (hazard concentration for a given percentage of species - HCps) to be estimated from the SSDs, and to be selected for the derivation of SSVs, with the adequate level of protection. It was proven that this selection may vary for different contaminants, however a clear justification should be given, in each case. The SSvs proposed in this study were for: U (151.4 mg U kg-1dw), Cd (5.6 mg Cd kg-1dw), and Cu (58.5 mg Cu kg-1dw) These values should now be tested for their descriminating power of soils with different levels of contamination. However, this studies clarifies the approach that should be followed for the derivation of SSVs for other metals and organic contaminants, and for other dominant types of Portuguese natural soils.

Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction $F$ on ${\mathbb{R}}^{n}$, we construct a probability measure on the set of all solutions to the Cauchy problem $\dot x\in F(x)$ with $x(0)=0$. With probability one, the derivatives of these random solutions take values within the set $ext F(x)$ of extreme points for a.e.~time $t$. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Geometric optimization on visibility problems: metaheuristic and exact solutions

Os problemas de visibilidade têm diversas aplicações a situações reais. Entre os mais conhecidos, e exaustivamente estudados, estão os que envolvem os conceitos de vigilância e ocultação em estruturas geométricas (problemas de vigilância e ocultação). Neste trabalho são estudados problemas de visibilidade em estruturas geométricas conhecidas como polígonos, uma vez que estes podem representar, de forma apropriada, muitos dos objectos reais e são de fácil manipulação computacional. O objectivo dos problemas de vigilância é a determinação do número mínimo de posições para a colocação de dispositivos num dado polígono, de modo a que estes dispositivos consigam “ver” a totalidade do polígono. Por outro lado, o objectivo dos problemas de ocultação é a determinação do número máximo de posições num dado polígono, de modo a que quaisquer duas posições não se consigam “ver”. Infelizmente, a maior parte dos problemas de visibilidade em polígonos são NP-difíceis, o que dá origem a duas linhas de investigação: o desenvolvimento de algoritmos que estabelecem soluções aproximadas e a determinação de soluções exactas para classes especiais de polígonos. Atendendo a estas duas linhas de investigação, o trabalho é dividido em duas partes. Na primeira parte são propostos algoritmos aproximados, baseados essencialmente em metaheurísticas e metaheurísticas híbridas, para resolver alguns problemas de visibilidade, tanto em polígonos arbitrários como ortogonais. Os problemas estudados são os seguintes: “Maximum Hidden Vertex Set problem”, “Minimum Vertex Guard Set problem”, “Minimum Vertex Floodlight Set problem” e “Minimum Vertex k-Modem Set problem”. São também desenvolvidos métodos que permitem determinar a razão de aproximação dos algoritmos propostos. Para cada problema são implementados os algoritmos apresentados e é realizado um estudo estatístico para estabelecer qual o algoritmo que obtém as melhores soluções num tempo razoável. Este estudo permite concluir que as metaheurísticas híbridas são, em geral, as melhores estratégias para resolver os problemas de visibilidade estudados. Na segunda parte desta dissertação são abordados os problemas “Minimum Vertex Guard Set”, “Maximum Hidden Set” e “Maximum Hidden Vertex Set”, onde são identificadas e estudadas algumas classes de polígonos para as quais são determinadas soluções exactas e/ou limites combinatórios.

Multiplicity results for some classes of Schrödinger-Poisson systems

In this thesis, we study the existence and multiplicity of solutions of the following class of Schr odinger-Poisson systems: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; where l 2 L2(R3) or l 2 L1(R3). And we consider that the nonlinearity satis es the following three kinds of cases: (i) a subcritical exponent with (x; u) = k(x)jujp 2u + h(x)u (4 p < 2 ) under an inde nite case; (ii) a general inde nite nonlinearity with (x; u) = k(x)g(u) + h(x)u; (iii) a critical growth exponent with (x; u) = k(x)juj2 2u + h(x)jujq 2u (2 q < 2 ). It is worth mentioning that the thesis contains three main innovations except overcoming several di culties, which are generated by the systems themselves. First, as an unknown referee said in his report, we are the rst authors concerning the existence of multiple positive solutions for Schr odinger- Poisson systems with an inde nite nonlinearity. Second, we nd an interesting phenomenon in Chapter 2 and Chapter 3 that we do not need the condition R R3 k(x)ep 1dx < 0 with an inde nite noncoercive case, where e1 is the rst eigenfunction of +id in H1(R3) with weight function h. A similar condition has been shown to be a su cient and necessary condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity for a bounded domain (see e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), or to be a su cient condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity in RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Moreover, the process used in this case can be applied to study other aspects of the Schr odinger-Poisson systems and it gives a way to study the Kirchho system and quasilinear Schr odinger system. Finally, to get sign changing solutions in Chapter 5, we follow the spirit of Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, but the procedure is simpler than that they have proposed in their paper.